Diffusion flames in a chamber

Gaseous fuel is injected at a constant rate M from the bottom of a narrow insulated chamber and oxidant diffuses from the top. The object is to determine the response curve, i.e. the unburnt fuel fraction R as a function of M for fixed Damköhler number (representing either the constant chamber pressure or the chamber's length). The conclusions we draw are based on activation-energy asymptotics. Explicit response curves are obtained over the whole range of D. Two critical values D_a, D₀ are uncovered and determined explicitly. If D < D_a the fuel will not burn: R ≡ 1 (extinguished state). If D_a < D < D₀ a closed curve, called an isola, is also a possible response; its base lies on R = 0 (complete burning) and its upper part corresponds to a partial burning of fuel. If D > D₀ the response is S-shaped: the top of the curve lies on R = 1 and extends to M → ∞; the bottom lies on R = 0 and persists however small M becomes. The isola has two extinction points, corresponding to the fuel being supplied too slowly or too fast to a flame at the bottom or top of the chamber. The S has just the latter, but also has an ignition point; as M is decreased through the corresponding value the flame suddenly appears. The asymptotic analysis completes the inadequate picture obtained previously by numerical methods. It spans the whole range of parameter values and enables explicit formulas to be written for the ignition and extinction points.